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We propose a coherent-form energy conservation relation (ECR) that is generally valid for the elastic transmission and reflection of a guided mode in a symmetric scattering system. In contrast with the classical incoherent-form ECR, |τ|2 + |ρ|2≤1 with τ and ρ denoting the elastic transmission and reflection coefficients of a guided mode, the coherent-form ECR is expressed as |τ + ρ|≤1, which imposes a constraint on a coherent superposition of the transmitted and reflected modes. The coherent-form ECR is rigorously demonstrated and is numerically tested by considering different types of modes in various scattering systems. Further discussions with the scattering matrix formalism indicate that two coherent-form ECRs, |τ + ρ|≤1 and |τ−ρ|≤1, along with the classical ECR |τ|2 + |ρ|2≤1 constitute a complete description of the energy conservation for the elastic scattering of a guided mode in a symmetric scattering system. The coherent-form ECR provides a common tool in terms of energy transfer for understanding and analyzing the scattering dynamics in currently interested scattering systems.
© 2013 Optical Society of America
1. Introduction
The energy conservation relation (ECR) of electromagnetic fields is a fundamental principle in electrodynamics and optics [1] and has significant resultant in the scattering theories of optical waveguides [2]. With recent advances in nano-optics and plasmonics, various subwavelength scattering systems are proposed that exhibit interesting properties such as the extraordinary optical transmission through subwavelength metallic hole arrays [3], negative refractive index in fishnet optical metamaterials [4], enhancement of nonlinear spectra by nano-antennas [5], and tunable cavities composed of subwavelength silica wires [6], which have important applications ranging from subwavelength imaging, bio-chemical sensing to light emitting. These fascinating properties have been shown related to the scattering of some guided modes, such as the surface plasmon polariton (SPP) in metallic waveguides [3,5,7,8] or the photonic modes in dielectric waveguides [6,9], which provide efficient channels for the energy transfer in these scattering systems. The ECR imposed on the scattering of guided modes has been shown playing an important role in understanding and analyzing the scattering dynamics of these systems [7,8,10,11].
As sketched by the insets in Fig. 1, the elastic scattering of a guided mode [12] at a scatterer can be characterized by a transmission coefficient τ and a reflection coefficient ρ. Classical ECR [2] has imposed a constraint of |τ|2 + |ρ|2≤1 on an incoherent superposition of τ and ρ. In this work, we propose that besides the classical ECR, τ and ρ also satisfy an ECR expressed as |τ + ρ|≤1, in which τ and ρ are superposed coherently. Rigorous demonstration of the coherent-form ECR is provided and its general validity is tested for different types of modes in various scattering systems. Besides imposing impact on classical theories of optical waveguides [2], the coherent-form ECR provides a common tool for analyzing the energy transfer via guided modes and the resultant properties of currently interested scattering systems, for instance, for deriving resonance conditions driven by the multiple scattering of some guided modes [13].
Fig. 1 Numerical test of the coherent-form ECR |τ + ρ|≤1 for different types of modes in various symmetric scattering systems. (a)-(b) Scattering of a SPP at a periodic chain of infinite-depth square holes (a) or at an infinite-depth slit (b) in gold substrate with an air cladding (refractive index 1). The chain period is 940nm, and the hole side length and the slit width are 266nm. (c)-(d) For the fundamental SPP mode on a gold nano-wire in air scattered at an air gap with a width of 30nm (c) and 100nm (d). The nano-wire has a square cross section with a side length of 40nm. (e)-(g) For the fundamental SPP mode in a gold/air/gold waveguide (air-gap width 100nm), which is scattered at an infinite-depth slit (slit width 50nm) (e), at a finite-depth groove (groove width 50nm and depth 1μm) (f), and at an orthogonal corner (g). (h) Coupling of the fundamental mode of two tightly contacted silica nano-wires (diameter 400nm, refractive index 1.46 at wavelength λ = 633nm) in air. The considered fundamental mode is polarized in the plane determined by the two wire axes.
2. Numerical test of the coherent-form energy conservation relation
To obtain a concrete knowledge of the coherent-form ECR, we first provide a variety of numerical results showing that the coherent-form ECR |τ + ρ|≤1 generally holds for various symmetric scattering systems and for different types of modes. Figures 1(a) and 1(b) present for different wavelengths the values of |τ + ρ| for the scattering of the SPP at two line scatterers, a periodic chain of infinite-depth holes [Fig. 1(a)] and an infinite-depth slit [Fig. 1(b)] in gold substrate, which are the elementary scatterers constituting the periodic metallic apertures supporting the extraordinary optical transmission [3,7,14]. The line scatterers are illuminated by a SPP at an air-gold interface with a unitary coefficient, which excites a transmitted SPP with a coefficient τ and a reflected SPP with a coefficient ρ (see the insets). The incident, transmitted and reflected SPP modes are assumed obeying the same normalization of fields. The elastic SPP scattering coefficients τ and ρ can be obtained as the scattering matrix elements with a fully-vectorial aperiodic Fourier modal method (a-FMM) [15]. Details of the calculation can be found in [7]. The wavelength-dependent refractive index of gold takes tabulated values from [16]. As shown in Fig. 1(a), the SPP transmission and reflection are almost energy conservative (|τ + ρ|≈1) as the wavelength is larger than the chain period (940nm), since now the transmitted and reflected higher-order surface waves (propagating nonperpendicularly to the chain that acts as a grating) are evanescent and do not carry energy. Also note that all the modes in the subwavelength holes are evanescent and do not carry energy [3,7]. While as the wavelength is smaller than the chain period, |τ + ρ| is distinctly smaller than one due to the appearance of higher-order propagative surface waves.
For the results shown in Fig. 1(b), the width of the subwavelength slit is chosen equal to the side length of the square holes in the chain. The intrinsic difference between the slit and the chain of holes is that the former always supports a fundamental SPP mode that is propagative [14]. This implies that for the slit, |τ + ρ| could be distinctly smaller than one due to the lost energy carried away by the propagative SPP mode in the slit, as confirmed by the results in Fig. 1(b).
Figures 1(c) and 1(d) provide the values of |τ + ρ| for the fundamental SPP mode on a gold nano-wire in air scattered by an air nano-gap. This represents an elementary scattering process in resonant nano-antennas for achieving a strong enhancement of field in the nano-gap [5]. It is seen that for all wavelengths, |τ + ρ| for the gap width of 100nm [Fig. 1(d)] is distinctly smaller than that for the gap width of 30nm [Fig. 1(c)], which is due to the higher energy loss scattered into free space at wider air gaps.
For the fundamental SPP mode in a metal-insulator-metal waveguide that may constitute compact plasmonic circuits due to its deep subwavelength confinement of the SPP mode [17], Figs. 1(e)-1(g) provide the values of |τ + ρ| for the scattering of the SPP mode at an infinite-depth slit, a finite-depth groove and an orthogonal corner, respectively. Note that for Fig. 1(g), for which the transmission waveguide is not collinear with the incidence waveguide, the coefficient τ of the transmitted SPP is extracted from the total field with the use of the mode orthogonality (see Eq. (1.36) in chapter 1.2.6 of [2]). Similar method has been used to extract the coefficient of the SPP mode at a single dielectric-metal interface [18]. For the calculation the τ is expressed as an overlap integral between the SPP mode and the total field (with the same expression as the α+ in Eq. (4) of [18] where the single-interface SPP field ESP and HSP should be replaced by the fundamental SPP mode). For the SPP scattering at the slit [Fig. 1(e)], |τ + ρ| is shown distinctly smaller than one for all wavelengths, which is due to the lost energy carried away by the propagative SPP mode in the slit. In comparison, |τ + ρ| for the SPP scattering at the groove [Fig. 1(f)] is distinctly smaller than one only at some specific wavelengths, and at other wavelengths, |τ + ρ| is quite close to unity. Further calculations show that the low values of |τ + ρ| are due to the Fabry-Perot resonance of the propagative SPP mode in the groove, which occurs at some specific wavelengths and which causes a considerable absorbance of energy by the lossy metal. For the results in Fig. 1(g), as expected, |τ + ρ| is quite close to one for all wavelengths since almost no energy is carried away or absorbed by channels other than the transmitted and the reflected SPP modes.
Besides plasmonic scattering systems, we also checked the validity of the coherent-form ECR for scattering systems of dielectric waveguides. As shown in Fig. 1(h), the coupling of the fundamental mode of two tightly contacted silica nano-wires is considered. Such silica nano-wires have shown their importance in constructing tunable micro optical systems such as micro-cavities for sensing or lasing [6]. For the calculation we take the same geometrical parameters as those in Fig. 2(b) (y polarized) in [9]. |τ + ρ| is shown quite close to one when a strong coupling occurs (|τ|≈1) for a coupling length L~2.5μm, which results in a weak scattering loss of energy into free space, while |τ + ρ| is distinctly smaller than one when the coupling is weak.
Fig. 2 Theoretical demonstration of the coherent-form ECR for a symmetric scattering system. As shown in (a), the considered guided mode with a unitary coefficient is transmitted with a coefficient τ and reflected with a coefficient ρ. (b) Related scattering problem for the demonstration, in which two identical modes with a unitary coefficient are incident coherently from both ports of the system. (c) General description for the elastic scattering of a guided mode in a symmetric scattering system.
3. Theoretical demonstration of the coherent-form energy conservation relation and further discussions with the scattering matrix formalism
The general validity of the ECR implies that there may exist a rigorous demonstration, which will be presented in the following to achieve a theoretical understanding. The considered symmetric scattering system is sketched in Fig. 2(a), which is composed of two identical waveguides (sketched by the horizontal solid lines) on both sides of a symmetric scatterer (sketched by the vertical dotted lines). The considered guided mode is incident from the left port with a unitary coefficient, which excites the same reflected mode (with a coefficient ρ) and transmitted mode (coefficient τ). Inspired by the coherent form of the ECR, for the demonstration we would rather consider another related scattering problem, in which two modes with unitary coefficients are incident coherently from both ports of the system, as shown in Fig. 2(b). This scattering problem can be treated as a coherent superposition of two original scattering problems shown in Fig. 1(a), as labeled with the green and red arrows, respectively. Thus for the related scattering problem, the coefficients of the two out-going modes are τ + ρ. With the use of the energy conservation principle of Maxwell’s equations for the total electromagnetic field, for Fig. 2(b) we can obtain,
which simply reduces to |τ + ρ|≤1. It should be noted that for deriving relation (1), the identical waveguides on both sides of the scatterer are assumed to be approximately lossless, so that the energy flux of the total field on both sides of the scatterer is approximately equal to the summation of the energy flux of every guided modes that constitute the total field [2]. This assumption is valid for lossless dielectric waveguides [such as the silica wire considered in Fig. 1(h)], or approximately valid for common plasmonic waveguides that work below the metal plasma frequency [such as those considered in Figs. 1(a)-1(g)], for which the field only penetrates slightly into lossy metals and thus experiences a weak energy loss.The theoretical demonstration proves that the coherent-form ECR is generally valid, provided that the scatterer is symmetric and that the waveguides on both sides of the scatterer are identical and approximately lossless.
To further check this general validity, Fig. 3 provides another example considering the transmission and reflection of a normally incident plane wave at a metallic hole array drilled in a gold membrane in air (see the inset). The transmission and reflection coefficients of the zeroth-order plane wave are denoted by t and r, respectively. Classical grating theories provide |t|2 + |r|2≤1. The coherent-form ECR predicts that t and r should also satisfy |t + r|≤1. This prediction is confirmed by the numerical results shown in Fig. 3, which are obtained with a fully-vectorial rigorous coupled wave analysis (RCWA) [19]. As shown in Fig. 3(a) for a real lossy metal of gold (taking refractive indices from [16]), |t + r|≤1 is satisfied over the whole considered range of wavelengths. There appears a sharp dip of |t + r| at a wavelength slightly larger than the array period (940nm), near which the extraordinary optical transmission occurs [3,7]. This dip is due to the energy loss resulting from a resonant excitation of surface waves on the membrane surface which causes the extraordinary optical transmission [7].
Fig. 3 Numerical test of the coherent-form ECR for the transmission (with a coefficient t) and reflection (coefficient r) of a normally incident plane wave at a metallic hole array drilled in a gold membrane in air (inset). The plane wave is polarized in one periodic direction of the grating. |t + r| for different wavelengths is shown. (a) For a real lossy metal of gold. (b) For an artificial lossless metal. The array period is 940nm, the side length of square holes is 266nm, and the membrane thickness is 200nm.
It may be interesting to examine the condition to achieve the equality of the coherent-form ECR. From the demonstration one can easily obtain the condition: for the waveguides on both sides of the scatterer, the transmitted and reflected guided mode is the sole mode that carries energy; within the region of the scatterer no energy is absorbed or carried away.
To confirm the analysis, we repeat the calculation of Fig. 3(a) but consider an artificial lossless metal. This is achieved by artificially removing the real part of the refractive index of the lossy gold, for instance, changing the gold refractive index of 0.25 + 6.84i (at λ = 1μm) to be 6.84i. The result is shown in Fig. 3(b). Now the dip of |t + r| disappears, and as the wavelength is larger than the array period of 940nm, |t + r| is exactly equal to 1 since now only the zeroth-order plane wave is propagative and all other higher-order plane waves are evanescent and do not carry energy. As the wavelength is smaller than the array period, |t + r| becomes smaller than 1 due to the appearance of higher-order propagative plane waves.
One may notice that in the demonstration of the ECR, the choice of the coherent superposition of the two incident modes is not unique. For instance, another choice by reversing the sign of one incident mode can give another coherent-form ECR, |τ −ρ|≤1, whose validity is also confirmed by our other numerical results not shown here. Now there arises a question: how many independent ECRs can one obtain at most for the elastic transmission and reflection of a guided mode? To answer the question, we consider a general description for the elastic scattering of a guided mode in a symmetric scattering system, as shown in Fig. 2(c). Two identical guided modes with arbitrary complex coefficients a1 and a2 are incident coherently from both sides of a symmetric scatterer, which excite two identical out-going modes with coefficients b1 and b2. The four coefficients are related by a scattering matrix S [2], b = Sa, with a = (a1,a2)T and b = (b1,b2)T being two column vectors, and
in view of Fig. 2(a). The sole relation for a and b that we can obtain from the energy conservation [2] is, bHb≤aHa for any a, where the superscript H means conjugate transpose of a complex matrix. With the use of b = Sa, this relation becomes aH(I−SHS)a≥0 for any a, where I represents an identity matrix. This is equivalent to say that I−SHS is a positive semidefinite matrix, which after some algebra derivations is further equivalent to three ECRs, |τ|2 + |ρ|2≤1, |τ + ρ|≤1 and |τ −ρ|≤1, no more than what we have obtained earlier. However, the statement here does not mean that other forms of ECRs [for instance, aH(I−SHS)a≥0 by selecting a = (1,−2i)T] are excluded, but means that all other forms of ECRs can be derived from the three ECRs.4. Conclusion
In summary, we have proposed a coherent-form ECR |τ + ρ|≤1 for the elastic transmission (with a coefficient τ) and reflection (coefficient ρ) of a guided mode in a symmetric scattering system. Rigorous demonstration of the ECR is provided showing that the ECR is generally valid for approximately lossless waveguides. Its general validity is numerically confirmed by considering different types of modes such as those in plasmonic waveguides, dielectric waveguides or free space and by considering different scattering systems. The proposed ECR is shown to reach the equality as the considered transmitted and reflected mode is the sole channel that carries away the energy. Further discussions with the scattering matrix formalism show that for the elastic scattering of a guided mode in a symmetric scattering system, what we can obtain at most from the energy conservation is no more than two coherent-form ECRs, |τ + ρ|≤1 and |τ−ρ|≤1, along with the classical incoherent-form ECR |τ|2 + |ρ|2≤1. The present results may be extended to the more general or complex cases, such as the asymmetric scattering systems, the inelastic scattering between modes of different types, or the scattering of Bloch modes in periodic waveguides [20].
Acknowledgments
Financial supports from the 973 Program (2013CB328701), the Natural Science Foundation of Tianjin (11JCZDJC15400) and the Natural Science Foundation of China (61322508) are acknowledged. The author thanks Dr. Philippe Lalanne for providing helpful comments.
References and notes
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